CN104318017A - Modeling method of asymmetric straight cylindrical gear pair - Google Patents

Abstract

The invention relates to a modeling method of an asymmetric straight gear pair, which comprises the steps of firstly calculating coordinates of equal-part key points on a tooth profile by using a tooth profile equation of an asymmetric straight gear, creating tooth profile key points by using ANSYS software, generating a tooth profile spline curve by using the key points, then establishing a single tooth end face, a whole tooth end face, an asymmetric straight gear entity model and an asymmetric straight gear transmission pair entity model from bottom to top, finally dividing a finite element grid, and setting dynamic constraint and load to obtain the asymmetric straight gear transmission pair finite element model. The whole modeling process of the invention realizes parameterization, and can greatly improve the finite element modeling quality of the gear transmission system and improve the analysis efficiency.

Description

A kind of modeling method of asymmetric cylindrical straight gear wheel set

Technical field

The present invention relates to a kind of gear finite element modeling method based on explicit dynamical Epidemiological Analysis, particularly the finite element modeling method of the asymmetric cylindrical straight gear wheel set of a kind of Double pressure angles.

Background technology

In single-direction transmission, increase flank of tooth compression-side pressure angle, make the compression-side flank of tooth form mal-distribution with not subjected to pressure lateral tooth flank, gear-driven registration can be significantly improved, reduce Gear Contact stress, Dedenda's bending stress, gear shaft constraint reaction, thus improve gear drive stability.Unsymmetric gear is the new gear of a kind of effective raising gear teeth performance and load-bearing capacity.The dynamic value simulation analysis of unsymmetric gear transmission is one of effective ways of its transmission performance of research and mechanism.Traditional gear modeling utilizes 3D sculpting software (Pro/E, UG), by the basic parameter of input standard spur gear wheel, set up the monodentate geometric model of symmetrical gear, then proceed in finite element analysis software and set up its finite element model, be further analyzed.Traditional modeling method is difficult to the Geometric Modeling realizing unsymmetric gear, geometric model proceeds in finite element software and is easy to lost part geological information, and each modeling can only be the gear Modling model of certain single particular type, increase and utilize finite element technique to study gear and the workload of design analysis.Traditional finite element analysis realizes engaging between single gear tooth or between monodentate with tooth bar mostly, usually based on the static analysis of many engagements during analysis, belongs to pseudo-performance analysis, can not realize the continuous dynamic simulation analysis in a complete swing circle of gear.

Summary of the invention

The technical problem to be solved in the present invention is to provide the finite element Direct Modeling of the asymmetric cylindrical straight gear wheel set of a kind of Double pressure angles based on explicit dynamical Epidemiological Analysis, the modeling efficiency of asymmetric Series Gear can be improved, achieve the parametrization of unsymmetric gear dynamic engagement analysis, intellectuality.

The technical scheme realizing the object of the invention is to provide a kind of modeling method of asymmetric cylindrical straight gear wheel set, comprises the steps:

1. parametrization sets up asymmetric involute profile curve, and detailed process is:

1.1) tooth curve of Unsymmetric involute gear is divided into: compression-side tooth curve, compression-side tooth root transition curve, not subjected to pressure side tooth curve, not subjected to pressure side tooth root transition curve four part; With the number of teeth of gear, modulus, gear compression-side pressure angle, not subjected to pressure wall pressure angle, addendum coefficient, tip clearance coefficient for input parameter, utilize the APDL programming language of ANSYS that asymmetric flank profil is carried out parametric programming, calculate tooth curve, every section of tooth curve is got a little by carrying out equal portions;

Gear compression-side flank profil key point coordinate (x _a, y _a):

$\left[\begin{array}{c}{x}_A\\ y_A\end{array}\right]=\frac{\mathrm{mn}\cos {\mathrm{\α}}_d}{2\cos {\mathrm{\α}}_{\mathrm{Ad}}}\left[\begin{array}{cc}-\sin (\mathrm{inv}{\mathrm{\α}}_d+\frac{\mathrm{\π}}{2n})& \cos (\mathrm{inv}{\mathrm{\α}}_d+\frac{\mathrm{\π}}{2n})\\ \cos (\mathrm{inv}{\mathrm{\α}}_d+\frac{\mathrm{\π}}{2n})& \sin (\mathrm{inv}{\mathrm{\α}}_d+\frac{\mathrm{\π}}{2n})\end{array}\right]\left[\begin{array}{c}\cos \left(\mathrm{inv}{\mathrm{\α}}_{\mathrm{Ad}}\right)\\ \sin \left(\mathrm{inv}{\mathrm{\α}}_{\mathrm{Ad}}\right)\end{array}\right]$

In above formula: α _adthe pressure angle of-compression-side curve point, $\tan {\mathrm{\α}}_d-\frac{4h_{\mathrm{ad}}^{*}}{z\·\sin \left(2{\mathrm{\α}}_d\right)}\≤{\mathrm{\α}}_{\mathrm{Md}}\≤\arccos \frac{z\·\cos {\mathrm{\α}}_d}{z+2h_{\mathrm{da}}^{*}};$

M-modulus; N-number of teeth; α _d-compression-side pressure angle of graduated circle; -compression-side addendum coefficient;

Gear compression-side tooth root key point coordinate (xB, yB), get a calculating by compression-side variable element equal portions within the scope of value value:

$\left[\begin{array}{c}{x}_B\\ y_B\end{array}\right]=\left[\begin{array}{cc}-\sin {\mathrm{\θ}}_d& \cos ({\mathrm{\α}}_{\mathrm{Bd}}-{\mathrm{\θ}}_d)\\ \cos {\mathrm{\θ}}_d& -\sin ({\mathrm{\α}}_{\mathrm{Bd}}-{\mathrm{\θ}}_d)\end{array}\right]\left[\begin{array}{c}\frac{\mathrm{mn}}{2}\\ R+\frac{Q}{\sin {\mathrm{\α}}_{\mathrm{Bd}}}\end{array}\right]$

In above formula: ${\mathrm{\θ}}_d=\frac{2(a\·\cot {\mathrm{\α}}_{\mathrm{Ad}}+\frac{\mathrm{\πm}}{2})}{\mathrm{mn}};$ α _bd-variable element, α _d≤ α _bd≤ 90 °;

R-to tip circle angular radius, $R=\frac{\frac{\mathrm{\πm}}{2}-({\tan \mathrm{\α}}_c+{\tan \mathrm{\α}}_d)(h_{\mathrm{ad}}^{*}+c_d^{*})m}{\sec {\mathrm{\α}}_c+\sec {\mathrm{\α}}_d-{\tan \mathrm{\α}}_c-{\tan \mathrm{\α}}_d};$

α _c-not subjected to pressure side pressure angle of graduated circle; -compression-side gear radial play coefficient;

Q-cutter parameters, $Q=(h_{\mathrm{ad}}^{*}+c_d^{*})m-R;$

Wheel not subjected to pressure side flank profil key point coordinate (x _c, y _c):

$\left[\begin{array}{c}{x}_C\\ y_C\end{array}\right]=\frac{\mathrm{mn}\cos {\mathrm{\α}}_c}{2\cos {\mathrm{\α}}_{\mathrm{Cc}}}\left[\begin{array}{cc}\sin (\mathrm{inv}{\mathrm{\α}}_c+\frac{\mathrm{\π}}{2n})& -\cos (\mathrm{inv}{\mathrm{\α}}_c+\frac{\mathrm{\π}}{2n})\\ \cos (\mathrm{inv}{\mathrm{\α}}_c+\frac{\mathrm{\π}}{2n})& \sin (\mathrm{inv}{\mathrm{\α}}_c+\frac{\mathrm{\π}}{2n})\end{array}\right]\left[\begin{array}{c}\cos \left(\mathrm{inv}{\mathrm{\α}}_{\mathrm{Cc}}\right)\\ \sin \left(\mathrm{inv}{\mathrm{\α}}_{\mathrm{Cc}}\right)\end{array}\right]$

In above formula: α _ccthe pressure angle of-not subjected to pressure side point,

$\arctan (\tan {\mathrm{\α}}_c-\frac{4h_{\mathrm{ad}}^{*}}{z\·\sin \left(2{\mathrm{\α}}_c\right)})\≤{\mathrm{\α}}_{\mathrm{Cc}}\≤\arccos \frac{z\·\cos {\mathrm{\α}}_c}{z+2h_{\mathrm{ad}}^{*}};$

Gear not subjected to pressure side tooth root key point coordinate (x _d, y _d):

$\left[\begin{array}{c}{x}_D\\ y_D\end{array}\right]=\left[\begin{array}{cc}-\sin {\mathrm{\θ}}_c& \cos ({\mathrm{\α}}_{\mathrm{Dc}}-{\mathrm{\θ}}_c)\\ \cos {\mathrm{\θ}}_c& -\sin ({\mathrm{\α}}_{\mathrm{Dc}}-{\mathrm{\θ}}_c)\end{array}\right]\left[\begin{array}{c}\frac{\mathrm{mn}}{2}\\ r_{\mathrm{\ρ}}+\frac{a}{\sin {\mathrm{\α}}_{\mathrm{Dc}}}\end{array}\right]$

In formula: ${\mathrm{\θ}}_c=\frac{2(Q\·\cot {\mathrm{\α}}_{\mathrm{Dc}}+\frac{\mathrm{\πm}}{2})}{\mathrm{mn}};$ α _dc-variable element, α _c≤ α _dc≤ 90 °;

1.2) adopt SPL to connect compression-side and not subjected to pressure side point respectively, generate tooth curve;

2. parametrization monodentate end face modeling, detailed process is:

2.1) cross tooth curve two peak, take true origin as the center of circle, made this circular arc of 2, generate tooth top circular arc;

2.2) generate a key point in true origin, connect two straight lines by true origin to two tooth root transition curve minimum points; Monodentate end face is generated by these two straight lines, compression-side tooth root transition curve, compression-side tooth curve, tooth top circular arc, not subjected to pressure side tooth curve, not subjected to pressure side tooth root transition curve;

3. parametrization whole tooth section modeling, detailed process is:

3.1) monodentate end face is copied number of teeth n monodentate under polar coordinates, deviation angle is 360/n degree successively;

3.2) superposition is carried out to each monodentate end face and obtain face of gear;

3.3) take true origin as the center of circle, justify with gear circular hole radius R K, by Boolean subtraction calculation, obtain gear face;

4. parametrization gear solid modelling, detailed process is:

Under cartesian coordinate system, by the length of gear face along end face vertical direction stretching facewidth B, obtain the three-dimensional entity model of gear;

5. parametrization follower gear modeling, detailed process is:

5.1) under cartesian coordinate system, gear face will move the distance of a gear pair center square;

5.2) 1., 2., 3., 4. set up the solid model of follower gear with step, if the driving gear number of teeth is even number, then follower gear rotates counterclockwise half follower gear monodentate angle around true origin;

6. parametrization gear finite element modeling, detailed process is:

6.1) gear material constant, cell type, cell parameters is defined;

6.2) flank of tooth unit size is set, adopts the division methods of sweeping grid, unsymmetric gear pair is divided finite element grid, two inner ring gears are divided into rigid element;

7. based on the unsymmetric gear contact of ANSYS explicit dynamical Epidemiological Analysis, constraint definition, definition load, detailed process is:

7.1) node group is created: select driving and driven gear compression-side node, and defined node group;

7.2) definition contact: on driving gear, node group is contact assembly, and on follower gear, node group is target element, arranges rubbing contact parameter;

7.3) barycenter of driving gear and follower gear is defined, the displacement of constraint driving gear and follower gear;

7.4) on driving gear inner ring, angular velocity is applied, locked-in torque on follower gear inner ring.

The present invention has positive effect: the present invention, by four sections of tooth profile curve equations of unsymmetric gear, joins as variable is got a little with the pressure angle of correspondence and change, adopts APDL Programming with Pascal Language, obtain tooth curve, avoids four sections of curves and occurs intersection and overlap.Generate unsymmetric gear solid model by tooth curve, definition explicit dynamical mathematic(al) parameter, can realize the dynamic engagement analysis of unsymmetric gear.By changing input parameter, the finite element model comprising the different parameters lower tooth wheel set of symmetrical gear can be built in modeling process.The present invention collects tooth Profile Design, solid modelling, parameter are defined in one, can realize parametrization that unsymmetric gear transmission dynamic engagement analyzes and intellectuality, effectively can put forward quality and the efficiency of finite element analysis.

Accompanying drawing explanation

The crucial point diagram of the asymmetric flank profil that Fig. 1 the present invention proposes;

The asymmetric tooth curve figure that Fig. 2 the present invention proposes;

The asymmetric monodentate end view drawing that Fig. 3 the present invention proposes;

The asymmetric whole tooth end view drawing that Fig. 4 the present invention proposes;

The straight driving gear sterogram of asymmetric cylinder that Fig. 5 the present invention proposes;

The asymmetric cylindrical straight gear wheel set sterogram that Fig. 6 the present invention proposes;

The asymmetric cylindrical straight gear wheel set finite element model figure that Fig. 7 the present invention proposes.

Embodiment

(embodiment 1)

Below in conjunction with the drawings and specific embodiments, set forth the present invention further, the modeling method of a kind of asymmetric cylindrical straight gear wheel set of the present embodiment comprises following several step:

1. parametrization sets up asymmetric involute profile curve:

Table 1: asymmetric spur gear wheel modeling parameters

Driving gear number of teeth n1	45	Follower gear number of teeth n2	22
				Modulus M	3mm	Facewidth B	22mm
Compression-side pressure angle AD	28°	Not subjected to pressure wall pressure angle AC	20°
				Addendum coefficient HA	1	Tip clearance coefficient CC	0.25

The tooth curve of Unsymmetric involute gear is divided into by 1.1: compression-side tooth curve, compression-side tooth root transition curve, not subjected to pressure side tooth curve, not subjected to pressure side tooth root transition curve four part.Be input parameter by the number of teeth of table 1 middle gear, modulus, gear compression-side pressure angle, not subjected to pressure wall pressure angle, addendum coefficient, tip clearance coefficient, utilize the APDL programming language of ANSYS that asymmetric tooth profile equation is carried out parametric programming, calculate tooth curve, every section of tooth curve is carried out 6 equal portions to be got a little, and number consecutively is 1 ~ 24.As shown in Figure 1.

Gear compression-side flank profil key point coordinate (xA, yA) (getting a calculating by compression-side pressure angle 6 equal portions within the scope of value value):

$\left[\begin{array}{c}{x}_A\\ y_A\end{array}\right]=\frac{\mathrm{mn}\cos {\mathrm{\α}}_d}{2\cos {\mathrm{\α}}_{\mathrm{Ad}}}\left[\begin{array}{cc}-\sin (\mathrm{inv}{\mathrm{\α}}_d+\frac{\mathrm{\π}}{2n})& \cos (\mathrm{inv}{\mathrm{\α}}_d+\frac{\mathrm{\π}}{2n})\\ \cos (\mathrm{inv}{\mathrm{\α}}_d+\frac{\mathrm{\π}}{2n})& \sin (\mathrm{inv}{\mathrm{\α}}_d+\frac{\mathrm{\π}}{2n})\end{array}\right]\left[\begin{array}{c}\cos \left(\mathrm{inv}{\mathrm{\α}}_{\mathrm{Ad}}\right)\\ \sin \left(\mathrm{inv}{\mathrm{\α}}_{\mathrm{Ad}}\right)\end{array}\right]$

In formula: α _adthe pressure angle (o) of-compression-side curve point,

$\tan {\mathrm{\α}}_d-\frac{4h_{\mathrm{ad}}^{*}}{z\·\sin \left(2{\mathrm{\α}}_d\right)}\≤{\mathrm{\α}}_{\mathrm{Md}}\≤\arccos \frac{z\·\cos {\mathrm{\α}}_d}{z+2h_{\mathrm{da}}^{*}},$

M-modulus (mm); N-number of teeth; α _d-compression-side pressure angle of graduated circle (o);

-compression-side addendum coefficient.

Gear compression-side tooth root key point coordinate (xB, yB) (getting a calculating by compression-side variable element 6 equal portions within the scope of value value):

$\left[\begin{array}{c}{x}_B\\ y_B\end{array}\right]=\left[\begin{array}{cc}-\sin {\mathrm{\θ}}_d& \cos ({\mathrm{\α}}_{\mathrm{Bd}}-{\mathrm{\θ}}_d)\\ \cos {\mathrm{\θ}}_d& -\sin ({\mathrm{\α}}_{\mathrm{Bd}}-{\mathrm{\θ}}_d)\end{array}\right]\left[\begin{array}{c}\frac{\mathrm{mn}}{2}\\ R+\frac{Q}{\sin {\mathrm{\α}}_{\mathrm{Bd}}}\end{array}\right]$

[in formula: ${\mathrm{\θ}}_d=\frac{2(a\·\cot {\mathrm{\α}}_{\mathrm{Ad}}+\frac{\mathrm{\πm}}{2})}{\mathrm{mn}};$ α _bd-variable element, α _d≤ α _bd≤ 90 °;

R-to tip circle angular radius (mm), $R=\frac{\frac{\mathrm{\πm}}{2}-({\tan \mathrm{\α}}_c+{\tan \mathrm{\α}}_d)(h_{\mathrm{ad}}^{*}+c_d^{*})m}{\sec {\mathrm{\α}}_c+\sec {\mathrm{\α}}_d-{\tan \mathrm{\α}}_c-{\tan \mathrm{\α}}_d},$

α _c-not subjected to pressure side pressure angle of graduated circle (o); -compression-side gear radial play coefficient;

Q-cutter parameters, $Q=(h_{\mathrm{ad}}^{*}+c_d^{*})m-R.$

Not subjected to pressure side flank profil key point coordinate (xC, yC) (getting a calculating by not subjected to pressure wall pressure angle 6 equal portions within the scope of value value): $\left[\begin{array}{c}{x}_C\\ y_C\end{array}\right]=\frac{\mathrm{mn}\cos {\mathrm{\α}}_c}{2\cos {\mathrm{\α}}_{\mathrm{Cc}}}\left[\begin{array}{cc}\sin (\mathrm{inv}{\mathrm{\α}}_c+\frac{\mathrm{\π}}{2n})& -\cos (\mathrm{inv}{\mathrm{\α}}_c+\frac{\mathrm{\π}}{2n})\\ \cos (\mathrm{inv}{\mathrm{\α}}_c+\frac{\mathrm{\π}}{2n})& \sin (\mathrm{inv}{\mathrm{\α}}_c+\frac{\mathrm{\π}}{2n})\end{array}\right]\left[\begin{array}{c}\cos \left(\mathrm{inv}{\mathrm{\α}}_{\mathrm{Cc}}\right)\\ \sin \left(\mathrm{inv}{\mathrm{\α}}_{\mathrm{Cc}}\right)\end{array}\right]$

In formula: α _ccthe pressure angle (o) of-not subjected to pressure side point,

$\arctan (\tan {\mathrm{\α}}_c-\frac{4h_{\mathrm{ad}}^{*}}{z\·\sin \left(2{\mathrm{\α}}_c\right)})\≤{\mathrm{\α}}_{\mathrm{Cc}}\≤\arccos \frac{z\·\cos {\mathrm{\α}}_c}{z+2h_{\mathrm{ad}}^{*}};$

Not subjected to pressure side tooth root key point coordinate (xD, yD) (getting a calculating by not subjected to pressure side variable element 6 equal portions within the scope of value value):

$\left[\begin{array}{c}{x}_D\\ y_D\end{array}\right]=\left[\begin{array}{cc}-\sin {\mathrm{\θ}}_c& \cos ({\mathrm{\α}}_{\mathrm{Dc}}-{\mathrm{\θ}}_c)\\ \cos {\mathrm{\θ}}_c& -\sin ({\mathrm{\α}}_{\mathrm{Dc}}-{\mathrm{\θ}}_c)\end{array}\right]\left[\begin{array}{c}\frac{\mathrm{mn}}{2}\\ r_{\mathrm{\ρ}}+\frac{a}{\sin {\mathrm{\α}}_{\mathrm{Dc}}}\end{array}\right]$

In formula: ${\mathrm{\θ}}_c=\frac{2(Q\·\cot {\mathrm{\α}}_{\mathrm{Dc}}+\frac{\mathrm{\πm}}{2})}{\mathrm{mn}};$ α _dc-variable element, α _c≤ α _dc≤ 90 °.

1.2) adopt SPL (BSPLIN) to connect compression-side and not subjected to pressure side point successively respectively, generate tooth curve L1, L2, as shown in Figure 2.

2. parametrization asymmetric monodentate end face modeling, comprising step has:

2.1) cross tooth curve two peak K12, K13, be the center of circle with true origin, made this circular arc of 2, generate tooth top circular arc L3.

2.2) generate a key point K25 in true origin, connect two straight lines L4, L5 by true origin to two tooth root transition curve minimum point K1, K24.The monodentate end face A1 shown in Fig. 3 is generated by L1, L2, L3, L4, L5.

3. parametrization asymmetric whole tooth section modeling, comprising step has:

3.1) monodentate end face is copied under polar coordinates tooth number Z 1 monodentate, and deviation angle is 360/Z1 degree successively.

3.2) superposition is carried out to each monodentate end face and obtain face of gear A2.

3.3) take true origin as the center of circle, make circle A4 with gear circular hole radius R K=40m, face of gear is subtracted computing to gear sky, obtains the gear face A5 shown in Fig. 4.

4. the asymmetric spur gear wheel solid modelling of parametrization: under cartesian coordinate system, by the length of gear face along end face vertical direction stretching facewidth B=22mm, obtains the gear three-dimensional entity model V1 shown in Fig. 5.

5. parametrization asymmetric cylinder follower gear modeling, comprising step has:

5.1) under cartesian coordinate system, gear face will move the distance of gear pair center square DY=M* (Z1+Z2)/2.

5.2) set up the solid model V2 of follower gear with step 1,2,3,4, follower gear model V2 is rotated counterclockwise half follower gear monodentate angle 180/Z2 around true origin, the asymmetric cylindrical straight gear wheel set solid model shown in Fig. 6 can be obtained.

6. parametrization asymmetric cylindrical straight gear wheel set finite element modeling, comprising step has:

6.1) the gear material constant shown in definition list 2, cell type.

Table 2: gear material parameter, cell type

6.2) arranging flank of tooth unit size is 2m, adopts the division methods of sweeping grid, asymmetric cylindrical straight gear wheel set is divided finite element grid SOLID164, two inner ring gears are divided into rigid element SHELL163.

7. based on the unsymmetric gear contact of ANSYS explicit dynamical Epidemiological Analysis, constraint definition, definition load, comprising step has:

7.1) select driving gear compression-side node, and defined node is node group ZHUJIECHU, select follower gear compression-side node, and defined node is node group CONGJIECHU.

7.2) definition contact: with node group ZHUJIECHU on driving gear for contact assembly, on follower gear, node group is CONGJIECHU target element, arranges friction factor F=0.1.

7.3) barycenter of driving gear and follower gear is defined in its gear circle centre position, constraint driving gear and follower gear are except all displacements except the rotation displacement ROTZ of Z.

7.4) on driving gear inner ring, angular velocity RBRZ=150.8rad/s is added, locked-in torque RBMZ=230Nm on follower gear inner ring.

The asymmetric cylindrical straight gear wheel set finite element model obtained as shown in Figure 7.

Claims (1)

1. a modeling method for asymmetric cylindrical straight gear wheel set, is characterized in that comprising the steps:

1. parametrization sets up asymmetric involute profile curve, and detailed process is:

1.1) tooth curve of Unsymmetric involute gear is divided into: compression-side tooth curve, compression-side tooth root transition curve, not subjected to pressure side tooth curve, not subjected to pressure side tooth root transition curve four part; With the number of teeth of gear, modulus, gear compression-side pressure angle, not subjected to pressure wall pressure angle, addendum coefficient, tip clearance coefficient for input parameter, utilize the APDL programming language of ANSYS that asymmetric flank profil is carried out parametric programming, calculate tooth curve, every section of tooth curve is got a little by carrying out equal portions;

Gear compression-side flank profil key point coordinate (x _a, y _a):

$\left[\begin{array}{c}{x}_A\\ y_A\end{array}\right]=\frac{\mathrm{mn}{\cos \mathrm{\α}}_d}{2{\cos \mathrm{\α}}_{\mathrm{Ad}}}\left[\begin{array}{cc}-\sin ({\mathrm{inv\α}}_d+\frac{\mathrm{\π}}{2n})& \cos ({\mathrm{inv\α}}_d+\frac{\mathrm{\π}}{2n})\\ \cos ({\mathrm{inv\α}}_d+\frac{\mathrm{\π}}{2n})& \sin ({\mathrm{inv\α}}_d+\frac{\mathrm{\π}}{2n})\end{array}\right]\left[\begin{array}{c}\cos \left({\mathrm{inv\α}}_{\mathrm{Ad}}\right)\\ \sin \left({\mathrm{inv\α}}_{\mathrm{Ad}}\right)\end{array}\right]$

In above formula: α _adthe pressure angle of-compression-side curve point, ${\tan \mathrm{\α}}_d-\frac{{4h}_{\mathrm{ad}}^{*}}{z\·\sin \left({2\mathrm{\α}}_d\right)}\≤{\mathrm{\α}}_{\mathrm{Md}}\≤\arccos \frac{z\·{\cos \mathrm{\α}}_d}{z+{2h}_{\mathrm{ad}}^{*}};$

M-modulus; N-number of teeth; α _d-compression-side pressure angle of graduated circle; -compression-side addendum coefficient;

Gear compression-side tooth root key point coordinate (x _b, y _b), get a calculating by compression-side variable element equal portions within the scope of value value:

$\left[\begin{array}{c}{x}_B\\ y_B\end{array}\right]=\left[\begin{array}{cc}{-\sin \mathrm{\θ}}_d& \cos ({\mathrm{\α}}_{\mathrm{Bd}}-{\mathrm{\θ}}_d)\\ {\cos \mathrm{\θ}}_d& -\sin ({\mathrm{\α}}_{\mathrm{Bd}}-{\mathrm{\θ}}_d)\end{array}\right]\left[\begin{array}{c}\frac{\mathrm{mn}}{2}\\ R+\frac{Q}{{\sin \mathrm{\α}}_{\mathrm{Bd}}}\end{array}\right]$

In above formula: ${\mathrm{\θ}}_d=\frac{2(a\·{\cot \mathrm{\α}}_{\mathrm{Ad}}+\frac{\mathrm{\πm}}{2})}{\mathrm{mn}};$ α _bd-variable element, α _d≤ α _bd≤ 90 °;

R-to tip circle angular radius, $R=\frac{\frac{\mathrm{\πm}}{2}-({\tan \mathrm{\α}}_c+{\tan \mathrm{\α}}_d)(h_{\mathrm{ad}}^{*}+c_d^{*})m}{{\sec \mathrm{\α}}_c+{\sec \mathrm{\α}}_d-{\tan \mathrm{\α}}_c-{\tan \mathrm{\α}}_d};$

α _c-not subjected to pressure side pressure angle of graduated circle; -compression-side gear radial play coefficient;

Q-cutter parameters, $Q=(h_{\mathrm{ad}}^{*}+c_d^{*})m-R;$

Gear not subjected to pressure side flank profil key point coordinate (x _c, y _c):

$\left[\begin{array}{c}{x}_C\\ y_C\end{array}\right]=\frac{\mathrm{mn}{\cos \mathrm{\α}}_c}{{2\cos \mathrm{\α}}_{\mathrm{Cc}}}\left[\begin{array}{cc}\sin ({\mathrm{inv\α}}_c+\frac{\mathrm{\π}}{2n})& -\cos ({\mathrm{inv\α}}_c+\frac{\mathrm{\π}}{2n})\\ \cos ({\mathrm{inv\α}}_c+\frac{\mathrm{\π}}{2n})& \sin ({\mathrm{inv\α}}_c+\frac{\mathrm{\π}}{2n})\end{array}\right]\left[\begin{array}{c}\cos \left({\mathrm{inv\α}}_{\mathrm{Cc}}\right)\\ \sin \left({\mathrm{inv\α}}_{\mathrm{Cc}}\right)\end{array}\right]$

In above formula: α _ccthe pressure angle of-not subjected to pressure side point,

$\arctan ({\tan \mathrm{\α}}_c-\frac{{4h}_{\mathrm{ad}}^{*}}{z\·\sin \left({2\mathrm{\α}}_c\right)})\≤{\mathrm{\α}}_{\mathrm{Cc}}\≤\arccos \frac{z\·{\cos \mathrm{\α}}_c}{z+{2h}_{\mathrm{ad}}^{*}};$

Gear not subjected to pressure side tooth root key point coordinate (x _d, y _d):

$\left[\begin{array}{c}{x}_D\\ y_D\end{array}\right]=\left[\begin{array}{cc}{-\sin \mathrm{\θ}}_c& \cos ({\mathrm{\α}}_{\mathrm{Dc}}-{\mathrm{\θ}}_c)\\ {\cos \mathrm{\θ}}_c& -\sin ({\mathrm{\α}}_{\mathrm{Dc}}-{\mathrm{\θ}}_c)\end{array}\right]\left[\begin{array}{c}\frac{\mathrm{mn}}{2}\\ r_{\mathrm{\ρ}}+\frac{a}{{\sin \mathrm{\α}}_{\mathrm{Dc}}}\end{array}\right]$

In formula: ${\mathrm{\θ}}_c=\frac{2(Q\·{\cot \mathrm{\α}}_{\mathrm{Dc}}+\frac{\mathrm{\πm}}{2})}{\mathrm{mn}};$ α _dc-variable element, α _c≤ α _dc≤ 90 °;

1.2) adopt SPL to connect compression-side and not subjected to pressure side point respectively, generate tooth curve;

2. parametrization monodentate end face modeling, detailed process is:

2.1) cross tooth curve two peak, take true origin as the center of circle, made this circular arc of 2, generate tooth top circular arc;

2.2) generate a key point in true origin, connect two straight lines by true origin to two tooth root transition curve minimum points; Monodentate end face is generated by these two straight lines, compression-side tooth root transition curve, compression-side tooth curve, tooth top circular arc, not subjected to pressure side tooth curve, not subjected to pressure side tooth root transition curve;

3. parametrization whole tooth section modeling, detailed process is:

3.1) monodentate end face is copied number of teeth n monodentate under polar coordinates, deviation angle is 360/n degree successively;

3.2) superposition is carried out to each monodentate end face and obtain face of gear;

3.3) take true origin as the center of circle, justify with gear circular hole radius R K, by Boolean subtraction calculation, obtain gear face;

4. parametrization gear solid modelling, detailed process is:

Under cartesian coordinate system, by the length of gear face along end face vertical direction stretching facewidth B, obtain the three-dimensional entity model of gear;

5. parametrization follower gear modeling, detailed process is:

5.1) under cartesian coordinate system, gear face will move the distance of a gear pair center square;

5.2) 1., 2., 3., 4. set up the solid model of follower gear with step, if the driving gear number of teeth is even number, then follower gear rotates counterclockwise half follower gear monodentate angle around true origin;

6. parametrization gear finite element modeling, detailed process is:

6.1) gear material constant, cell type, cell parameters is defined;

6.2) flank of tooth unit size is set, adopts the division methods of sweeping grid, unsymmetric gear pair is divided finite element grid, two inner ring gears are divided into rigid element;

7. based on the unsymmetric gear contact of ANSYS explicit dynamical Epidemiological Analysis, constraint definition, definition load, detailed process is:

7.1) node group is created: select driving and driven gear compression-side node, and defined node group;

7.2) definition contact: on driving gear, node group is contact assembly, and on follower gear, node group is target element, arranges rubbing contact parameter;

7.3) barycenter of driving gear and follower gear is defined, the displacement of constraint driving gear and follower gear;

7.4) on driving gear inner ring, angular velocity is applied, locked-in torque on follower gear inner ring.